Problem 44691. Comparison of floating-point numbers (singles)

Created by David Verrelli

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{"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","targetMode":"","relationshipId":"rId1","target":"/matlab/document.xml"},{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/output","targetMode":"","relationshipId":"rId2","target":"/matlab/output.xml"}],"parts":[{"partUri":"/matlab/document.xml","relationship":[],"contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/matlab_prog/floating-point-numbers.html\"><w:r><w:t>Floating-point numbers</w:t></w:r></w:hyperlink><w:r><w:t> cannot generally be represented exactly, so it is usually inappropriate to test for 'equality' between two floating-point numbers. Rather, it is generally appropriate to check whether the difference between the two numbers is sufficiently small that they can be considered practically equal. (In other words, so close in value that any differences could be explained by inherent limitations of computations using floating-point numbers.)</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Based on two scalar inputs of type</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/ref/double.html\"><w:r><w:t>double</w:t></w:r></w:hyperlink><w:r><w:t>, namely</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>A</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>B</w:t></w:r><w:r><w:t>, you must return a scalar</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/ref/logical.html\"><w:r><w:t>logical</w:t></w:r></w:hyperlink><w:r><w:t> output set to</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/ref/true.html\"><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>true</w:t></w:r></w:hyperlink><w:r><w:t> if the difference in magnitude between</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>the</w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/ref/single.html\"><w:r><w:rPr><w:b/></w:rPr><w:t>single</w:t></w:r></w:hyperlink><w:r><w:rPr><w:b/></w:rPr><w:t>-precision representations</w:t></w:r><w:r><w:t> of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>A</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>B</w:t></w:r><w:r><w:t> is 'small', or otherwise set to</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/ref/false.html\"><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>false</w:t></w:r></w:hyperlink><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>For</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>this</w:t></w:r><w:r><w:t> problem \"small\" shall mean no more than</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>Δ</w:t></w:r><w:r><w:t>, which is defined as the larger of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>δ</w:t></w:r><w:r><w:t> and ten times</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>ε</w:t></w:r><w:r><w:t>. In turn,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>δ</w:t></w:r><w:r><w:t> has a fixed value of 1×10⁻¹⁰ and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>ε</w:t></w:r><w:r><w:t> is the larger of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>ε</w:t></w:r><w:r><w:t>₁ &amp;</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>ε</w:t></w:r><w:r><w:t>₂, where</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>ε</w:t></w:r><w:r><w:t>₁ is the</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://au.mathworks.com/help/matlab/ref/eps.html\"><w:r><w:t>floating-point precision</w:t></w:r></w:hyperlink><w:r><w:t> with which</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>A</w:t></w:r><w:r><w:t> is represented, and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>ε</w:t></w:r><w:r><w:t>₂ is the precision with which</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>B</w:t></w:r><w:r><w:t> is represented.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>EXAMPLE:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[% Input\nA = 0\nB = 1E-20\n\n% Output\npracticallyEqual = true]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Explanation: When represented as</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>single</w:t></w:r><w:r><w:t> values,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>A</w:t></w:r><w:r><w:t> is represented with a precision of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>ε</w:t></w:r><w:r><w:t>₁ = 2⁻¹⁴⁹, whereas</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>B</w:t></w:r><w:r><w:t> is represented with a precision of</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>ε</w:t></w:r><w:r><w:t>₂ = 2⁻⁹⁰. Thus</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>ε</w:t></w:r><w:r><w:t> = 2⁻⁹⁰; however the threshold</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>Δ</w:t></w:r><w:r><w:t> is 1×10⁻¹⁰ for this situation, because 1×10⁻¹⁰ &gt; 10×2⁻⁹⁰. The difference between</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>A</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>B</w:t></w:r><w:r><w:t> is 1×10⁻²⁰, and this difference does not exceed the threshold. Thus</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>A</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>B</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>are</w:t></w:r><w:r><w:t> practically equal in this example.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>RELATED PROBLEMS:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/44690\"><w:r><w:t>Problem 44690: Comparison of floating-point numbers (doubles)</w:t></w:r></w:hyperlink></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/44631\"><w:r><w:t>Problem 44631: Make your own Test Suite (part 0)</w:t></w:r></w:hyperlink></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/44513\"><w:r><w:t>Problem 44513: Add all the numbers between two limits (inclusive)</w:t></w:r></w:hyperlink></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

<a href = "https://au.mathworks.com/help/matlab/matlab_prog/floating-point-numbers.html">Floating-point numbers</a> cannot generally be represented exactly, so it is usually inappropriate to test for 'equality' between two floating-point numbers. Rather, it is generally appropriate to check whether the difference between the two numbers is sufficiently small that they can be considered practically equal. (In other words, so close in value that any differences could be explained by inherent limitations of computations using floating-point numbers.)Based on two scalar inputs of type <a href = "https://au.mathworks.com/help/matlab/ref/double.html">double</a>, namely <tt>A</tt> and <tt>B</tt>, you must return a scalar <a href = "https://au.mathworks.com/help/matlab/ref/logical.html">logical</a> output set to <a href = "https://au.mathworks.com/help/matlab/ref/true.html"><tt>true</tt></a> if the difference in magnitude between the <a href = "https://au.mathworks.com/help/matlab/ref/single.html">single</a>-precision representations of <tt>A</tt> and <tt>B</tt> is 'small', or otherwise set to <a href = "https://au.mathworks.com/help/matlab/ref/false.html"><tt>false</tt></a>.For this problem "small" shall mean no more than Δ, which is defined as the larger of δ and ten times ε. In turn, δ has a fixed value of 1×10⁻¹⁰ and ε is the larger of ε₁ & ε₂, where ε₁ is the <a href = "https://au.mathworks.com/help/matlab/ref/eps.html">floating-point precision</a> with which <tt>A</tt> is represented, and ε₂ is the precision with which <tt>B</tt> is represented.EXAMPLE:<pre class="language-matlab">% Input
A = 0
B = 1E-20
</pre><pre class="language-matlab">% Output
practicallyEqual = true
</pre>Explanation: When represented as <tt>single</tt> values, <tt>A</tt> is represented with a precision of ε₁ = 2⁻¹⁴⁹, whereas <tt>B</tt> is represented with a precision of ε₂ = 2⁻⁹⁰. Thus ε = 2⁻⁹⁰; however the threshold Δ is 1×10⁻¹⁰ for this situation, because 1×10⁻¹⁰ &gt; 10×2⁻⁹⁰. The difference between <tt>A</tt> and <tt>B</tt> is 1×10⁻²⁰, and this difference does not exceed the threshold. Thus <tt>A</tt> and <tt>B</tt> are practically equal in this example.RELATED PROBLEMS:<ul><li><a href = "https://www.mathworks.com/matlabcentral/cody/problems/44690">Problem 44690: Comparison of floating-point numbers (doubles)</a></li><li><a href = "https://www.mathworks.com/matlabcentral/cody/problems/44631">Problem 44631: Make your own Test Suite (part 0)</a></li><li><a href = "https://www.mathworks.com/matlabcentral/cody/problems/44513">Problem 44513: Add all the numbers between two limits (inclusive)</a></li></ul>

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David Verrelli on 12 Aug 2018

Oops. I just noticed that I forgot to rename the function from "compareDoubles" (used in Problem 44690) to "compareSingles". Unfortunately I cannot make the fix now without breaking the two successful submissions. Sorry about any confusion. —DIV

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